Core Odoo pros: Odoo is great at what it does. It's modular, and let's you consolidate a lot of business software into one place where it can all talk to itself. Generally, this makes accounting teams very happy, since invoicing, inventory (esp GAAP), and expenses are all generated mostly automatically. The UI has really improved recently, and performance scales well. Odoo has some good functional tutorials on their website that guide you through using the features.
Core Odoo cons: If your business has some need outside of what Odoo expects you to use it for, good luck. Unless it's a minor thing, you're probably better off changing your SOPs to fit Odoo rather than trying to change Odoo to work for you. Of course, you don't know whether it's a minor thing unless you have someone with Odoo familiarity -- but we'll get to that.
Custom Odoo pros: Almost all of the business logic is written in Python with a fully extensible framework. It's designed to be modified, and it's pretty friendly. The docs aren't great, but the community is. The Odoo Community Association (OCA) maintains a large library of OS extensions, divided by business area. Even if you don't find what you need, it's a great place to learn by example on creating custom modules. Simple UI changes are mainly handled by XML, which again is fairly easy to pick up by studying examples.
Custom Odoo cons: There's a learning curve, as with anything. Something you thought should be a simple change might turn out to be a pit of vipers. If you're trying to do web app changes, I will pray for you. Odoo has started to roll out OWL, a JS framework based on React and Node.js, but documentation on JS in Odoo is basically non-existent, and Odoo devs avoid JS like the plague.
If your company is looking to change, Odoo has implementation partners, who will tailor Odoo to your needs. Partner prices vary widely, and you get what you pay for. It really depends on your company's size and budget. You still pay licensing, etc. through Odoo itself.
Avoid odoo.sh (Odoo's SaaS option) at all costs, I have never seen it go well. Avoid Studio (allows UI changes without coding) at all costs, it turns into spaghetti every time.